When the explicit MPC method fails to apply to a problem due to scalability or nonlinearity, one may consider another alternative, namely, function approximation.
Function approximation approach implements a MPC policy by approximating offline that MPC control law with an efficient surrogate model, such as a neural network, and uses that surrogate model for online deployment.
Early works in this direction include [1]}, [2]}, which has described a procedure to design a surrogate NN controller and has proposed conditions to guarantee its closed-loop stability.
Later development has focused on guaranteeing different theoretical properties of the NN-controlled closed-loop system.
For example, [3]} proposes a NN method with probabilistic optimality bounds;
[4]} has designed a projection operator that can modify any arbitrary NN controller to a stabilizing one;
[5]} proposes a mixed-integer linear program to certify stability and feasibility of ReLU-activated NN controllers;
and [6]} has developed a method that guarantees certain performance indicator of a NN-controlled system using probabilistic validation methods.
Besides, rather than replacing the original MPC controller, a very recent work [7]} uses a NN to warm start an optimization solver, which can then solve the MPC policy faster in real time.
For a comprehensive study, readers may consult [8]}.
Applications of function approximation methods have found successes in practice, such as [9]}.
